I am a physicist and I am pondering over a particular generalization of Stokes' theorem and Maxwell's equations. They apply to vector fields like the electric or magnetic one. However if the vectors aren't true elements of $R^3$ but elements of a Lie group, say SO(3) or SU(2), I wonder how to formulate Stokes' theorem (or Maxwell's equations). The derivative of a Lie group would be an element of the Lie algebra, thus one would put two incompatible quantities into one equation.

Let me add one more explanation why there should be a solution to the problem: For small values, elements of SO(3) (threedimensional rotations) correspond to "vectors": express the
rotation axis as direction of the vector and the twisting angle
as its length (the Euler axis representation). But for finite
rotations, this does not work. Mind also that finite rotations
do not commute, while vector addition does.

The correspondence between "small" elements of $SO_3$ and vectors is called the exponential map for the Lie group. This is the function that sends a skew-adjoint matrix $A$ (3x3 in your case, so its determined by three real numbers) to $I + A + A^2/2 + A^3/3! + \cdots $. Provided $A$ is small, addition of matrices corresponds to matrix multiplication, at least, to first order. To second order you start seeing Lie brackets appear.
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Ryan BudneyMar 6 '13 at 16:42